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Theorem imass1 6097
Description: Subset theorem for image. (Contributed by NM, 16-Mar-2004.)
Assertion
Ref Expression
imass1 (𝐴𝐵 → (𝐴𝐶) ⊆ (𝐵𝐶))

Proof of Theorem imass1
StepHypRef Expression
1 ssres 6006 . . 3 (𝐴𝐵 → (𝐴𝐶) ⊆ (𝐵𝐶))
2 rnss 5936 . . 3 ((𝐴𝐶) ⊆ (𝐵𝐶) → ran (𝐴𝐶) ⊆ ran (𝐵𝐶))
31, 2syl 17 . 2 (𝐴𝐵 → ran (𝐴𝐶) ⊆ ran (𝐵𝐶))
4 df-ima 5688 . 2 (𝐴𝐶) = ran (𝐴𝐶)
5 df-ima 5688 . 2 (𝐵𝐶) = ran (𝐵𝐶)
63, 4, 53sstr4g 4026 1 (𝐴𝐵 → (𝐴𝐶) ⊆ (𝐵𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wss 3947  ran crn 5676  cres 5677  cima 5678
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-rab 3434  df-v 3477  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-br 5148  df-opab 5210  df-cnv 5683  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688
This theorem is referenced by:  predrelss  6335  vdwnnlem1  16924  dprdres  19890  imasnopn  23176  imasncld  23177  imasncls  23178  utoptop  23721  restutop  23724  ustuqtop3  23730  utopreg  23739  metustbl  24057  imadifxp  31810  esum2d  33029  eulerpartlemmf  33312  bj-imdirco  36009  brtrclfv2  42411  frege97d  42436  frege109d  42441  frege131d  42448  hess  42464  resimass  43877  setrecsss  47648
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